It's only now that I found out that Saitama's class exists.
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Im not gonna tell you the entire thing ill just say that if you keep taking on the biggest of challenges and training till your body cant even move. you'll be proven worthy of the secret character (The secret character is sometimes insanely op)
But if you still are stuck and dont care about spoilers then the way to unlock is beneath this.
In order to unlock this character you need to play on the hardest difficulty (Which is unlocked by raising wage at the end of the game.) And beat the final boss. Then you must show how menacing you are by stomping on his face until he is dead. Then you'll be given a new option instead of raising wage called "Hit the gym" It will teleport you to a new level where there are mats that you have to interact with 100 times each. Then Break both of the windows and run to the side of the screen and you'll be teleported to the other side then go past a poster saying how many times you have passed it. Do that 10 times then you'll be proven worthy of the secret character...